When memorizing the kanji lessons, focus on the reading they teach you.
If I understand correctly your problem seems to be on vocabulary items. On a personal level, I don’t think the way Wanikani teaches vocab is that great, because it encourages people too much to focus on the component kanji of the word instead of the pronunciation, which is the most important.
For example let’s think of 一月. Many people on WK seem to approach items like this in the following manner when they come up in reviews.
“Ok so it’s one + moon, I learned that as January. Now for the reading… hmmm one was いち, and moon… had too readings, which one was it again? Dammit let’s try いちげつ. Ah shoot, it was wrong.”
That approach never worked for me, and I was endlessly frustrated, until I changed the way I learned vocab to actually focus on the pronunciation first. My approach would be:
“Ok, let’s try to sound out the kanji and see if it matches any words I know. Do I know any いちげつ? Nope, doesn’t ring a bell. Do I know any いちがつ? Oh yeah, that’s January, I know that one.”
I use a reorder script for reviews so that I always get prompted for the reading first, and then the meaning, in order to facilitate this learning process.
Exceptions
Of course some vocab is just weird because it doesn’t follow the kanji readings at all. and I’ll have to learn them a bit differently, like 今日 or 昨日, but this method helps me in over 80% of cases. Even in these rare cases, I focus first on burning the meaning + reading in my head first, and then attach kanji to it as a secondary process. When speaking to people it’s less important to remember the kanji than it is to remember the pronunciation, so I focus on reading first.
Nerdy details about what can and can't be on'yomi
As for remembering if a reading is kun or on, if you already know the vocab words it’s easy enough to tell which is which. For one, if the vocab word has hiragana attached that’s part of the pronunciation, like in 食べる, 早い, it’s a safe bet that that’s the kun’yomi reading. If the reading is very short and usually part of multi-kanji words, like 事故, that’s a safe bet that it’s on’yomi. But there are exceptions to this “rule” so it isn’t reliable for all cases.
However, there are certain combinations of kana that can’t possibly be a on’yomi. For example, on’yomi can’t be more than 2 mora long. As for the second mora, it’s usually an vowel to extend the sound in the first mora (い for the ‘e’ sound, like 映、う for the ‘o’ sound, like in 公, or う again for the ‘u’ sound, like in 数), or one of these: く, き, つ and ん. If you find a reading that’s two mora, and doesn’t end in the above-mentioned sounds, it’s not a on’yomi reading. Similarly, readings that are three or more mora can’t be on’yomi.
You can read more about it on Wikipedia: Kanji - Wikipedia